Central projection drawing. Drawing

To transition from a spatial representation of an object to its flat image, the projection method is used.

In order for a three-dimensional object located in three-dimensional space to be “transferred” to a plane, i.e., to obtain its image, it is necessary to project it. To do this, from a point in space selected in a certain way, which is called the center of projection, it is necessary to draw straight lines (rays) through each point of the depicted object. These lines are called projecting lines. The plane on which we received the image of the object is called the projection plane, and the image of the object that we receive on this plane is called its projection.

Depending on the position of the center of projection and the direction of the projecting rays in relation to the projection plane, the projection can be either central (conical) or parallel (cylindrical).

The most common case of obtaining projections of spatial figures is the central projection.

In this case, the projecting rays come out from one point - the projection center S, which is at a finite distance from the projection plane P 1.

In order to obtain the central projections of points A And B, it is necessary to draw projecting rays from the projection center S through points A And B until it intersects with the projection plane P 1. When intersecting, points are obtained A 1 And B 1— central projections of points A And B.

Point position S and planes P 1, which does not pass through the center of the projections, determine the central projection apparatus. If it is given, then it is always possible to determine the position of the central projection of any point in space onto the projection plane, and each point in space will have only one central projection. However, from one central projection it is impossible to determine the position of a point in space, since it can be located anywhere on a straight line connecting the projection of the point and the center of the projection.

To determine the position of a point A in space according to its central projections, it is necessary to have two central projections of this point A 1 And A 2, obtained from two different centers S 1 And S 2. If we draw projecting rays S 1 A 1 And S 2 A 2, then the point of their intersection will uniquely determine the position of the point A in space.

To construct a central projection A 1 B 1 segment AB it is enough to construct central projections A 1 And B 1 points A And IN, since two points uniquely define a line.

Central projection is highly visual, as it corresponds to the visual perception of objects.

Properties of projections with central projection:

The projection of a point is a point.
The projection of a line is a line.
In general, the projection of a line is a straight line. (If the straight line coincides with the projecting ray, then its projection is a point).
If a point belongs to a line, then the projection of the point belongs to the projection of the line.
The point of intersection of the lines is projected to the point of intersection of the projections of these lines.
In general, a planar polyhedron is projected into a polyhedron with the same number of vertices.
The projection of mutually parallel lines is a pencil of lines.
If a plane figure is parallel to the plane of projections, then its projection is similar to this figure.

The projection of point A onto the projection plane π 1 is the point A 1 of intersection of the projecting line with the projection plane π 1 passing through point A (Fig. 1.1):

The projection of any geometric figure is the set of projections of all its points. The direction of the projecting straight lines and the position of the π 1 planes determine the projection apparatus.

Central projection is a projection in which all projecting rays emanate from one point S - the center of projection (Fig. 1.2).

Parallel projection is a projection in which all projecting lines are parallel to a given direction S (Fig. 1.3).

Rice. 1.1. Projection of point A onto the projection plane π 1

Rice. 1.2. Center projection example

Rice. 1.3. Parallel Projection Example

Parallel projection is a special case of central projection, when the point S is at an infinite distance from the projection plane π 1.

With a given projection apparatus, each point in space corresponds to one and only one point on the projection plane.

One projection of a point does not determine the position of this point in space. Indeed, the projection A 1 can correspond to an infinite number of points A ’, A ’’, ... located on the projecting line (Fig. 1.4).

To determine the position of a point in space with any projection apparatus, it is necessary to have two of its projections, obtained with two different projection directions (or with two different projection centers).

Rice. 1.4. An example of the location of a set of points on a projecting line

So, from Fig. 1.5 it is clear that two projections of point A (A 1 and A 2), obtained with two directions of projection S 1 and S 2, uniquely determine the position of point A itself in space - as the intersection of projecting lines 1 and 2 drawn from projections A 1 and A 2 parallel to the projection directions S 1 and S 2 .

Rice. 1.5. Determining the position of point A in space

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7.1. Manifold geometric shapes in nature. In mathematics lessons you have already become acquainted with some geometric figures. A figure is any collection (set) of points. All sorts of complex figure can be divided into simpler ones.

If all the points of a figure lie in the same plane, the figure is called flat: triangle, square, etc. A set of points located in space forms a spatial figure: cube, cylinder, etc. Figures in space are called bodies.

The objects that surround us, machine parts, as a rule, have a complex real geometric shape. However, if you look closely at them, you will notice that some of them consist of one or more simple geometric bodies or their modified parts. Such geometric bodies that form the shape of objects are prisms (Fig. 22, a), pyramids (Fig. 22, b), cylinders (Fig. 23, a), cones (Fig. 23, b), balls, etc.

Rice. 22

Rice. 23

The shape of each geometric body has its own characteristic features. By them we distinguish a prism from a cylinder, a pyramid from a cone, etc. These signs are also used when constructing drawings of geometric bodies or objects and parts consisting of them. However, before making such drawings, let’s find out what rules underlie the methods of their construction.

What geometric bodies do you know?
Look around you and find simple geometric bodies in the shape of surrounding objects.

7.2. General information about projection. Images of objects on drawings in accordance with the rules state standard performed using the method (method) of rectangular projection. By projection we will call the process of obtaining projections of an object.

Let's look at an example. Let's take an arbitrary point A and some plane in space. Let us denote this plane, for example, by the capital letter P (pi) of the Greek alphabet with index one - i.e. P 1 (Fig. 24). Let's draw a straight line through point A so that it intersects the plane P 1 at some point A." Then point A" will be the projection of point A. We will denote the projections of points with the same letters as the points themselves, but with a prime. The plane on which the projection is obtained is called projection plane. Direct AA" is called projecting beam. With its help, point A is projected onto the plane P 1.

Rice. 24

Note. There are other designations for projections of points - A 1, A 2, A 3 - and projection planes - H, V, W.

Using this method, projections of all points of any figure can be constructed. So, in order to obtain the projection A "B" of the straight segment AB (Fig. 25, a), it is necessary to draw projecting rays through two points of the segment - A and B. Moreover, if the straight line or its segment coincides in direction with the projecting ray (segment CD in Fig. 25, b), they are projected onto the projection plane to a point. In the images, the projections of matching points are indicated by the sign =, for example: C = D", as in Figure 25, b.

Rice. 25

To construct a projection of a figure, it is necessary to draw imaginary projecting rays through its points until they intersect with the projection plane. The projections of all points of a figure on the plane form the projection of a given figure.

Consider, for example, obtaining a projection of such a geometric figure as a triangle (Fig. 26).

Rice. 26

Projection of point A onto given plane P 1 will be point A" as a result of the intersection of the projecting ray AA" with the projection plane. The projections of points B and C will be points B" and C. By connecting points A", B" and C on the plane with straight segments, we obtain the figure A" B "C, which will be the projection of the given figure.

In what follows, by the term projection we will understand the image of an object on a projection plane.

The word "projection" is Latin. Translated, it means “throw (throw) forward.”

Place a flat object on the paper and trace it with a pencil. You will receive an image corresponding to the projection of this object. Examples of projections are photographs, film frames, etc.

Images of objects obtained by projection will be called projection images.

What is projection?
How to construct a projection of a point on a plane? projection of the figure?

7.3. Center and parallel projection. If the projecting rays, with the help of which the projection of an object is constructed, come from one point, the projection is called central (Fig. 27). The point from which the rays originate is called projection center. The resulting projection is called central.

Rice. 27

The central projection is often called perspective. Examples of central projection are photographs and film frames, shadows cast from an object by the rays of an electric light bulb, etc. Central projections are used in drawing from life.

If the projecting rays are parallel to each other (Fig. 28), then the projection is called parallel, and the resulting projection is parallel. Parallel projection can be conditionally considered to be the solar shadows of objects. Examples of parallel projection are shown in Figures 25, a and 26.

Rice. 28

It is easier to construct an image of an object with parallel projection than with central projection. So, if segment AB (Fig. 28) or any flat figure, as, for example, in Fig. 29, are parallel to the projection plane, then their projections and the projected figures themselves are equal.

Rice. 29

With parallel projection, all rays fall on the projection plane at the same angle. If this is any angle that is not equal to 90°, as in Figure 29, a or Figure 25, a, then the projection is called oblique.

In the case when the projecting rays are perpendicular to the projection plane (see Fig. 29, b), i.e., they make an angle of 90° with it, projection is called rectangular(see Fig. 26). The resulting projection is called rectangular.

What projection is called central? parallel? oblique? rectangular?
Why is it easier to construct an image in a parallel projection than in a central one?

7.4. Obtaining axonometric projections. In technical graphics, a special group consists of projections that are obtained by parallel projection of an object along with the x, y and z axes of the spatial system of rectangular coordinates onto an arbitrary plane (Fig. 30). Let's denote it P 0 . The projection obtained in this way on the plane P 0 is called axonometric. Depending on the direction of projection in relation to the projection plane, axonometric projections can be either rectangular or oblique.

Rice. thirty

The word "axonometry" is Greek. Translated, it means “measurement along the axes.”

Projections of x 0, y 0 and z 0 coordinate axes on the projection plane are called axonometric. When constructing axonometric projections of objects, dimensions are laid along the axes or parallel to them.

Axonometric projections are classified as visual images. You can easily get them general idea about the external form of an object.

However, on axonometric projections, objects are distorted. For example, circles are projected into ellipses, right angles into obtuse or acute angles. Some dimensions of the object are also distorted. Therefore, such projections are used mainly when performing technical drawings.

To obtain images in drawings, the method of rectangular projection onto one, two or more projection planes is used.

What projections are called axonometric?
What axonometric projections are obtained depending on the direction of projection?

2) *if the projecting rays are perpendicular to the projection plane

3) if the projecting rays come from one point

4) if the projecting rays are directed in different directions

What is the central projection sometimes called?

1) oblique

2) *perspective

3) rectangular

4) parallel

10. The plane located in front of the viewer is called:

1) horizontal

2) profile

3) *front

4) central

What projection is called central?

1) if the projecting rays are parallel to each other

2) *if the projecting rays come from one point

3) if the projecting rays are perpendicular

4) if the projecting rays diverge

What is a section called?

1) projection of a figure obtained by intersecting an object with a plane

2) *image of a figure obtained by intersecting an object with a plane

3) display of a figure obtained by intersecting an object with a plane

4) geometric figure, obtained by connection

13. Which image is called a section:

1) *image of an object mentally dissected by a plane

2) display the figure

3) projection of an object mentally dissected by a plane

4) image of a figure connected to a plane

Which incision is called local?

1) *cut to show internal structure part of the part we need

2) a cut allowing to show external structure details

3) a cut allowing to show half of the part

4) a section made along the plane of symmetry of the part

What line in the drawings separates part of the view and part of the section?

1) dashed line

2) thick line

3) thin line

4) *dash-dotted line

16. Rectangular isometric projection is performed in axes located at angles to each other:

1) *120, 120, 120 degrees

2) 135, 135, 90 degrees

3) 180, 90, 90 degrees

4) 130, 130, 100 degrees

17. What ruler is used to draw an ellipse:

1) tire

2) *patterns

3) square

4) protractor

18. As a result of the intersection of a cone with a plane parallel to its base, we obtain:

1) truncated pyramid

2) truncated triangle

3) *truncated cone

4) truncated circle

19. A body formed by rotating a circle around one of its diameters is called:

1) triangle

2) cone

4) ellipse

20. According to GOST 2.312-72 symbol means:

1) seam along a closed contour

2) *seam with removed reinforcement

3) interrupted seam with staggered sections

4) a seam that has a smooth transition to the base metal

B5. Electrical Engineering with Fundamentals of Industrial Electronics

What law is the operating principle of welding transformers based on?

1) *by law electromagnetic induction

2) on Ohm's law, where I=U/R

3) on the law of the magnetic circuit

4) based on Kirchhoff's law

Which transformers allow you to smoothly change the voltage at the output terminals?

1) power transformers

2) instrument transformers

3) autotransformers

4) *welding transformers

3. Electronic devices that convert direct voltage into alternating voltage are called:

1) rectifiers

2) *inverters

3) converters

4) transformers

What current is called constant?

1) current varying in magnitude and direction

2) *current does not change in magnitude and direction

3) current varying in magnitude

4) current changing in direction

Introduction

All sections descriptive geometry use one method - the projection method, therefore drawings used not only in descriptive geometry are called projection drawings.

The projection method is that any of the points of a set of points in space can be projected using projecting rays onto any surface. To do this, imagine some given surface (Fig. 1) and a point A in space. When carrying out the beam S through the point A in the direction of the surface the latter will intersect it at the point A 1 . Full stop A called projected point. The plane α on which the projection is obtained is called projection plane. The point of intersection of the ray with the plane is called the projection of the point A. Straight AA 1 (beam), called projecting beam.

Fig.1.

The central (conical or polar) projection method is based on the fact that when projecting a series of points onto a plane ( A, B, C etc.) all projecting rays pass through one point called projection center, or pole.

Let's imagine a triangle in space ABC and projecting rays passing through a given pole S and through the points ABC triangles drawn to the intersection with the plane α. Triangle A 1 B 1 C 1 will be the central projection of the triangle ABC(Fig. 2).

The central projection method does not satisfy a number of conditions necessary for a technical drawing, namely: it does not provide a uniform image, complete clarity of all geometric shapes, does not have easy measurability, and does not have simplicity of image.

The method of parallel (oblique) projection is that all projecting rays passing through the points of the triangle ABC, will be parallel to each other (Fig. 3). This method follows from the method of central projection, in which the pole must be removed at an infinitely large distance from the plane onto which the object is projected.

Orthogonal (rectangular) projection method is a method when the projecting rays are parallel to each other and perpendicular to the projection plane (Fig. 4). This method– a special case of parallel projection.

Thus, any point in space can be projected onto the projection plane: horizontal P 1, frontal P 2 and profile P 3. The horizontal projection of a point is indicated A 1 or A′, front A 2 or A″, profile A 3 or A′″ (Fig. 5).